Center of a graph
= Center of a graph
{title2=$C(G)=\operatorname*{arg\,min}_{v\in V(G)}e(v)$}
The <center of a graph> is the set of vertices with minimum <graph eccentricity>. For a finite <connected graph> $G$, put
$$
e(v)=\max_{u\in V(G)}d_G(v,u),\qquad C(G)=\{v:e(v)=\operatorname{rad}G\}.
$$
Here $d_G$ is <shortest-path distance> and $\operatorname{rad}G=\min_v e(v)$ is the <graph radius>. The center solves a facility-placement problem in which the cost is the distance to the farthest vertex.
In a five-vertex <path graph>, eccentricities are $4,3,2,3,4$, so only the middle vertex is central. In a four-vertex path, the two middle vertices both have eccentricity two. The <center of a tree> is always a single vertex or a pair of adjacent vertices: repeatedly deleting all leaves exposes its middle. For a general graph the center need not have this form; every vertex of a <cycle graph> is central.
Minimizing the sum of distances defines a different objective. For example, let $a,b,c,d$ form a path, and attach three additional leaves to $a$. The center is $\{b\}$, but the sum of distances is smallest at $a$. The farthest-distance criterion and the average-distance criterion can therefore select different locations.